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- 1. Chapter 6 Continuous Probability Distributions Uniform Probability Distribution Normal Probability Distribution Exponential Probability Distribution f(x) x µ© 2003 South-Western/Thomson Learning™ Slide 1
- 2. Continuous Probability Distributions A continuous random variable can assume any value in an interval on the real line or in a collection of intervals. It is not possible to talk about the probability of the random variable assuming a particular value. Instead, we talk about the probability of the random variable assuming a value within a given interval. The probability of the random variable assuming a value within some given interval from x1 to x2 is defined to be the area under the graph of the probability density function between x1 and x2.© 2003 South-Western/Thomson Learning™ Slide 2
- 3. Uniform Probability Distribution A random variable is uniformly distributed whenever the probability is proportional to the interval’s length. Uniform Probability Density Function f(x) = 1/(b - a) for a < x < b =0 elsewhere where: a = smallest value the variable can assume b = largest value the variable can assume© 2003 South-Western/Thomson Learning™ Slide 3
- 4. Uniform Probability Distribution Expected Value of x E(x) = (a + b)/2 Variance of x Var(x) = (b - a)2/12 where: a = smallest value the variable can assume b = largest value the variable can assume© 2003 South-Western/Thomson Learning™ Slide 4
- 5. Example: Slaters Buffet Uniform Probability Distribution Slater customers are charged for the amount of salad they take. Sampling suggests that the amount of salad taken is uniformly distributed between 5 ounces and 15 ounces. The probability density function is f(x) = 1/10 for 5 < x < 15 =0 elsewhere where: x = salad plate filling weight© 2003 South-Western/Thomson Learning™ Slide 5
- 6. Example: Slaters Buffet Uniform Probability Distribution What is the probability that a customer will take between 12 and 15 ounces of salad? f(x) P(12 < x < 15) = 1/10(3) = .3 1/10 x 5 10 12 15 Salad Weight (oz.)© 2003 South-Western/Thomson Learning™ Slide 6
- 7. Example: Slaters Buffet Expected Value of x E(x) = (a + b)/2 = (5 + 15)/2 = 10 Variance of x Var(x) = (b - a)2/12 = (15 – 5)2/12 = 8.33© 2003 South-Western/Thomson Learning™ Slide 7
- 8. Normal Probability Distribution Graph of the Normal Probability Density Function f(x) x µ© 2003 South-Western/Thomson Learning™ Slide 8
- 9. Normal Probability Distribution Characteristics of the Normal Probability Distribution • The shape of the normal curve is often illustrated as a bell-shaped curve. • Two parameters, µ (mean) and σ (standard deviation), determine the location and shape of the distribution. • The highest point on the normal curve is at the mean, which is also the median and mode. • The mean can be any numerical value: negative, zero, or positive. … continued© 2003 South-Western/Thomson Learning™ Slide 9
- 10. Normal Probability Distribution Characteristics of the Normal Probability Distribution • The normal curve is symmetric. • The standard deviation determines the width of the curve: larger values result in wider, flatter curves. • The total area under the curve is 1 (.5 to the left of the mean and .5 to the right). • Probabilities for the normal random variable are given by areas under the curve.© 2003 South-Western/Thomson Learning™ Slide 10
- 11. Normal Probability Distribution Percent (%) of Values in Some Commonly Used Intervals • 68.26% of values of a normal random variable are within +/- 1 standard deviation of its mean. • 95.44% of values of a normal random variable are within +/- 2 standard deviations of its mean. • 99.72% of values of a normal random variable are within +/- 3 standard deviations of its mean.© 2003 South-Western/Thomson Learning™ Slide 11
- 12. Normal Probability Distribution Normal Probability Density Function 1 − ( x − µ )2 / 2σ2 f ( x) = e 2 πσ where: µ = mean σ = standard deviation π = 3.14159 e = 2.71828© 2003 South-Western/Thomson Learning™ Slide 12
- 13. Standard Normal Probability Distribution A random variable that has a normal distribution with a mean of zero and a standard deviation of one is said to have a standard normal probability distribution. The letter z is commonly used to designate this normal random variable. Converting to the Standard Normal Distribution x−µ z= σ We can think of z as a measure of the number of standard deviations x is from µ.© 2003 South-Western/Thomson Learning™ Slide 13
- 14. Using Excel to Compute Standard Normal Probabilities Excel has two functions for computing probabilities and z values for a standard normal distribution: • NORMSDIST is used to compute the cumulative probability given a z value. • NORMSINV is used to compute the z value given a cumulative probability. (The letter S in the above function names reminds us that they relate to the standard normal probability distribution.)© 2003 South-Western/Thomson Learning™ Slide 14
- 15. Using Excel to Compute Standard Normal Probabilities Formula Worksheet A B 1 Probabilities: Standard Normal Distribution 2 3 P (z < 1.00) =NORMSDIST(1) 4 P (0.00 < z < 1.00) =NORMSDIST(1)-NORMSDIST(0) 5 P (0.00 < z < 1.25) =NORMSDIST(1.25)-NORMSDIST(0) 6 P (-1.00 < z < 1.00) =NORMSDIST(1)-NORMSDIST(-1) 7 P (z > 1.58) =1-NORMSDIST(1.58) 8 P (z < -0.50) =NORMSDIST(-0.5) 9© 2003 South-Western/Thomson Learning™ Slide 15
- 16. Using Excel to Compute Standard Normal Probabilities Value Worksheet A B 1 Probabilities: Standard Normal Distribution 2 3 P (z < 1.00) 0.8413 4 P (0.00 < z < 1.00) 0.3413 5 P (0.00 < z < 1.25) 0.3944 6 P (-1.00 < z < 1.00) 0.6827 7 P (z > 1.58) 0.0571 8 P (z < -0.50) 0.3085 9© 2003 South-Western/Thomson Learning™ Slide 16
- 17. Using Excel to Compute Standard Normal Probabilities Formula Worksheet A B 1 Finding z Values, Given Probabilities 2 3 z value with .10 in upper tail =NORMSINV(0.9) 4 z value with .025 in upper tail =NORMSINV(0.975) 5 z value with .025 in lower tail =NORMSINV(0.025) 6© 2003 South-Western/Thomson Learning™ Slide 17
- 18. Using Excel to Compute Standard Normal Probabilities Value Worksheet A B 1 Finding z Values, Given Probabilities 2 3 z value with .10 in upper tail 1.28 4 z value with .025 in upper tail 1.96 5 z value with .025 in lower tail -1.96 6© 2003 South-Western/Thomson Learning™ Slide 18
- 19. Example: Pep Zone Standard Normal Probability Distribution Pep Zone sells auto parts and supplies including apopular multi-grade motor oil. When the stock of thisoil drops to 20 gallons, a replenishment order is placed. The store manager is concerned that sales are beinglost due to stockouts while waiting for an order. It hasbeen determined that leadtime demand is normallydistributed with a mean of 15 gallons and a standarddeviation of 6 gallons. The manager would like to know the probability of astockout, P(x > 20).© 2003 South-Western/Thomson Learning™ Slide 19
- 20. Example: Pep Zone Standard Normal Probability Distribution The Standard Normal table shows an area of .2967 for the region between the z = 0 and z = .83 lines below. The shaded tail area is .5 - .2967 = .2033. The probability of a stock- out is .2033. z = (x - µ)/σ Area = .2967 = (20 - 15)/6 = .83 Area = .5 - .2967 = .2033 Area = .5 z 0 .83© 2003 South-Western/Thomson Learning™ Slide 20
- 21. Example: Pep Zone Using the Standard Normal Probability Table z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 .0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359 .1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0753 .2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141 .3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517 .4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879 .5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224 .6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2518 .2549 .7 .2580 .2612 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852 .8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133 .9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389© 2003 South-Western/Thomson Learning™ Slide 21
- 22. Example: Pep Zone Standard Normal Probability Distribution If the manager of Pep Zone wants the probability of a stockout to be no more than .05, what should the reorder point be? Area = .05 Area = .5 Area = .45 0 z.05Let z.05 represent the z value cutting the .05 tail area.© 2003 South-Western/Thomson Learning™ Slide 22
- 23. Example: Pep Zone Using the Standard Normal Probability Table We now look-up the .4500 area in the Standard Normal Probability table to find the corresponding z.05 value. z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 . . . . . . . . . . . 1.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .4441 1.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545 1.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4633 1.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4706 1.9 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .4767 . . . . . . . . . . . z.05 = 1.645 is a reasonable estimate.© 2003 South-Western/Thomson Learning™ Slide 23
- 24. Example: Pep Zone Standard Normal Probability Distribution The corresponding value of x is given by x = µ + z.05σ = 15 + 1.645(6) = 24.87 A reorder point of 24.87 gallons will place the probability of a stockout during leadtime at .05. Perhaps Pep Zone should set the reorder point at 25 gallons to keep the probability under .05.© 2003 South-Western/Thomson Learning™ Slide 24
- 25. Using Excel to Compute Normal Probabilities Excel has two functions for computing cumulative probabilities and x values for any normal distribution: • NORMDIST is used to compute the cumulative probability given an x value. • NORMINV is used to compute the x value given a cumulative probability.© 2003 South-Western/Thomson Learning™ Slide 25
- 26. Using Excel to Compute Normal Probabilities Formula Worksheet for Pep Zone Example A B 1 Probabilities: Normal Distribution 2 3 P (x > 20) =1-NORMDIST(20,15,6,TRUE) 4 5 Finding x Values, Given Probabilities 6 7 x value with .05 in upper tail =NORMINV(0.95,15,6) 8© 2003 South-Western/Thomson Learning™ Slide 26
- 27. Using Excel to Compute Normal Probabilities Value Worksheet for Pep Zone Example A B 1 Probabilities: Normal Distribution 2 3 P (x > 20) 0.2023 4 5 Finding x Values, Given Probabilities 6 7 x value with .05 in upper tail 24.87 8 Note: P(x > 20) = .2023 here using Excel, while our previous manual approach using the z table yielded .2033 due to our rounding of the z value.© 2003 South-Western/Thomson Learning™ Slide 27
- 28. Exponential Probability Distribution Exponential Probability Density Function 1 − x /µ f ( x) = e for x > 0, µ > 0 µ where: µ = mean e = 2.71828© 2003 South-Western/Thomson Learning™ Slide 28
- 29. Exponential Probability Distribution Cumulative Exponential Distribution Function P ( x ≤ x0 ) = 1 − e − xo / µ where: x0 = some specific value of x© 2003 South-Western/Thomson Learning™ Slide 29
- 30. Using Excel to Compute Exponential Probabilities Excel’s EXPONDIST function can be used to compute exponential probabilities. The function has three arguments: • First – the value of the random variable x • Second – 1/µ (the inverse of the mean number of occurrences in an interval) • Third – “TRUE” or “FALSE” (we will always enter “TRUE” because we’re seeking a cumulative probability)© 2003 South-Western/Thomson Learning™ Slide 30
- 31. Using Excel to Compute Exponential Probabilities Formula Worksheet A B1 Probabilities: Exponential Distribution23 P (x < 18) =EXPONDIST(18,1/15,TRUE)4 P (6 < x < 18) =EXPONDIST(18,1/15,TRUE)-EXPONDIST(6,1/15,TRUE)5 P (x > 8) =1-EXPONDIST(8,1/15,TRUE)6© 2003 South-Western/Thomson Learning™ Slide 31
- 32. Using Excel to Compute Exponential Probabilities Value Worksheet A B1 Probabilities: Exponential Distribution23 P (x < 18) 0.69884 P (6 < x < 18) 0.36915 P (x > 8) 0.58666© 2003 South-Western/Thomson Learning™ Slide 32
- 33. Example: Al’s Carwash Exponential Probability Distribution The time between arrivals of cars at Al’s Carwash follows an exponential probability distribution with a mean time between arrivals of 3 minutes. Al would like to know the probability that the time between two successive arrivals will be 2 minutes or less. P(x < 2) = 1 - 2.71828-2/3 = 1 - .5134 = .4866© 2003 South-Western/Thomson Learning™ Slide 33
- 34. Example: Al’s Carwash Graph of the Probability Density Function f(x) .4 .3 P(x < 2) = area = .4866 .2 .1 x 1 2 3 4 5 6 7 8 9 10 Time Between Successive Arrivals (mins.)© 2003 South-Western/Thomson Learning™ Slide 34
- 35. Using Excel to Compute Exponential Probabilities Formula Worksheet for Al’s Carwash Example A B 1 Probabilities: Exponential Distribution 2 3 P (x < 2) =EXPONDIST(2,1/3,TRUE) 4© 2003 South-Western/Thomson Learning™ Slide 35
- 36. Using Excel to Compute Exponential Probabilities Value Worksheet for Al’s Carwash Example A B 1 Probabilities: Exponential Distribution 2 3 P (x < 2) 0.4866 4© 2003 South-Western/Thomson Learning™ Slide 36
- 37. Relationship between the Poisson and Exponential Distributions (If) the Poisson distribution provides an appropriate description of the number of occurrences per interval (If) the exponential distribution provides an appropriate description of the length of the interval between occurrences© 2003 South-Western/Thomson Learning™ Slide 37

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